Homework #4

1. Problem #1 on the previous homework was meant to be an easy application of the characterization of Dedekind domains we did in class. However, I overlooked the crucial point of the problem, which is show that the integral closure of R in L is Noetherian. If you figured out a way around that, then great, but if you were unable to do it before, here is an outline of an easier proof of this fac, but only in the case that L is a separable extension of K. We will recall the following factoids from Galois theory:

Factoid: Let Tr denote the trace of the field extension L/K, which is a K-linear map from L to K that sends l \in L to the sum of its conjugates by the distinct embeddings L into the algebraic closure of K. By a theorem from field theory, the trace map is nonzero when the extension is separable. You may also assume you know the following fact: For any basis {u_1,...,u_n} of L over K, there exists another basis {v_1,...,v_n} of L over K such that Tr(u_iv_j) = \delta_{ij} (the Kronecker delta).

a) Show that Tr(s) \in R for all s \in S. (Hint: Show it is integral over R)
b) Show that the fraction field of S is L.
c) Let {u_1,...,u_n} be a basis for L over K. We may assume that the u_i are in S since we can clear denominators. Let {v_1,...,v_n} be the dual basis from the factoid. Show that S \subseteq Rv_1 + ... + Rv_n. (Hint: use a)
d) Conclude that S is Noetherian and that S is hence a Dedekind domain.

All graded rings below are assumed to be nonnegatively graded over Artinian base R_0

2. Let R be a Noetherian graded ring and M a finitely generated graded R-module. Then for any k >= 1, M_n \subseteq (R_+)^k M for n sufficiently large.

3. Let R be a Noetherian graded ring with R_0 artinian, and M a finitely generated graded R-module. An element x \in R_l is superficial for M if (0:_M x) has finite length (i.e. (0:_M x)_n = 0 for all but finitely many n). Show that for every M f.g., there exists an element x \in R_l that is superficial for M. (Hint: Use graded primary decomposition, as well as the graded version of prime avoidance to show that you can pick an element x \in R_+ homogeneous that is not in any of the associated primes of M. Then show that x is superficial.)

4. Let R be a Noetherian graded ring, and M a f.g. graded R-module. Let x \in R_n be a nonzerodivisor on M. Compute H(M/xM,t) in terms of H(M,t).

5. Let R be a Noetherian graded ring and M a finitely generated R-module. Show that if x is a homogeneous nonzerodivisor of M, then dim M/xM = dim M - 1.

6. Let R be a Noetherian graded ring and M a f.g. graded R-module of dimension d. Suppose that x_1,...,x_d is a regular sequence on M generated in degree 1. Show that the H(M,t)(1-t)^d is a polynomial with nonnegative coefficients.

7. Prove number 5 in the local case as well.

8. Let (R,m) be a Noetherian local ring and set G = gr^m(R) (the associated graded ring of R with respect to the m-adic filtration). Show that if G is a domain, then so is R. Show that the converse can fail.

This homework is due on March 30th (the Tuesday after spring break).